Giant g-factors and fully spin-polarized states in metamorphic short-period InAsSb/InSb superlattices

Realizing a large Landé g-factor of electrons in solid-state materials has long been thought of as a rewarding task as it can trigger abundant immediate applications in spintronics and quantum computing. Here, by using metamorphic InAsSb/InSb superlattices (SLs), we demonstrate an unprecedented high value of g ≈ 104, twice larger than that in bulk InSb, and fully spin-polarized states at low magnetic fields. In addition, we show that the g-factor can be tuned on demand from 20 to 110 via varying the SL period. The key ingredients of such a wide tunability are the wavefunction mixing and overlap between the electron and hole states, which have drawn little attention in prior studies. Our work not only establishes metamorphic InAsSb/InSb as a promising and competitive material platform for future quantum devices but also provides a new route toward g-factor engineering in semiconductor structures.


SUPPLEMENTARY NOTE 2. EIGHT-BAND k · p CALCULATION
To correctly interpret the magneto-infrared spectroscopy data, we perform the k · p calculations [1][2][3][4][5]. We break down the total Hamiltonian into three parts which take into account the material band structure [4], the strain effect [5], and the Zeeman effect [3], respectively. The H k·p Hamiltonian near Γ point for zinc-blende type crystals is well documented. We employ the following basis Here, is the reduced Planck constant, k = (k x , k y , k z ) is the wave vector, k ± = k x ± ik y , band gap, ∆ is the split-off band gap, γ 1 ,γ 2 ,γ 3 are the modified Luttinger parameters, m 0 is the free electron mass, and P 0 is related to the Kane energy E p by E p = 2m 0 P 2 0 / 2 . Also, A c is related to the electron effective mass m * by A c = 2 /2m * −E p (3E g +2∆)/6m 0 E g (E g +∆).
In our calculation, we define the SL growth direction as the z direction. While k x and k y are still good quantum numbers due to the in-plane translational symmetry, we need to consider both the quantum confinement effect and the additional minibands from the thin film and periodic SL structure along the z direction. Therefore, we replace k z with k z = −i∂/∂ z + k ζ , where the momentum operator arises from the quantum confinement effect and k ζ stands for the minibands caused by the periodic SL structure. Specifically, k ζ takes a value between −π/t and π/t, where t is the period of the SL. In the presence of a magnetic field (B), we focus on the LL spectrum at k ζ = 0 since this is the van Hove singularity point where dominant optical absorption occurs.
Next, we consider the strain effect, which plays an important role in semiconductor structures. The strain Hamiltonian H s takes the following form Here, a c , a v , b, and d are deformation potentials. We apply the pseudomorphic approximation by assuming the lattice constants in the core structures throughout the SL pinned to the lattice constants of the virtual substrate. Then, the strain tensor reads where a 0 and a are the pinned and original lattice constants, respectively, and C 11 and C 12 are both the stiffness constants.
In the presence of a magnetic field, we also include the Zeeman effect using , and l c is the magnetic length. Supplementary Tab. 1 summarizes the parameters used in our eight-band calculations, and Supplementary Tab. 2 summarizes the strain parameters. The parameters for InSb in Supplementary Tabs. 1 and 2 are taken from Ref. [6]. For 52% Sb alloy, we use the interpolation scheme recommended in Ref. [6] while leaving E p and E v as fitting parameters to match the experimental data [7,8].
Supplementary Tab. 1: Parameters for band structure calculations.

Sb
E In all calculations, we use a piecewise function to describe the change of band parameters across the interface, and for simplicity, we take the axial approximation by replacing γ 2 and γ 3 with their average. In addition, we have rescaled the Kane energy E p so that A c = 0 to avoid spurious solutions [9].
Lastly, it is worth noting that for the split-off band gap ∆ in InAs 0.48 Sb 0.52 , we consider the bowing effect [6] with a positive bowing parameter. Though both a positive [7,10,11] and negative [12][13][14] bowing parameter have been reported in the literature, we choose a positive value to be self-consistent with that in Ref. [11] measured in similar SL samples.
We also find that because the split-off band is far away in energy and only affects the band structure through perturbation, the choice of bowing parameter for ∆ has a negligible effect on the model fitting in the main text.

SUPPLEMENTARY NOTE 3. SPIN POLARIZATION CALCULATION
Based on the above basis functions, we can calculate the spin polarization of the mth LL by summing up the corresponding spin components in the wavefunction spinor where ↑↓ denotes the spin directions, and f m k is the coefficient of the kth component in the eigenstates of the mth LL.

SUPPLEMENTARY NOTE 4. EXTENDED FITTING BETWEEN THE k·p MODEL AND THE FOUR-BAND MODEL
In Fig. 5(a,c) of the main text, we fit the low-field k ·p LLs with the effective Hamiltonian of the four-band model for two ultra-narrow band gap SLs, and good agreement between the two models is achieved. In Supplementary Fig. 2, we show an additional fitting for the case of InAs 0.48 Sb 0.52 /InSb (2-nm/1.12-nm) SL, where the band gap is relatively large E g ≈ 103 meV and in the ultrashort period limit. Great agreement between the two models is also evidenced.
To better understand the origin of the LL (spin) splitting in InAsSb/InSb SLs, we extend the fitting to a moderate high magnetic field. We consider a SL in the normal regime with a period of 3-nm/1.69-nm (sample 1444) as an example. Supplementary Fig. 3 shows the comparison between the k · p results and the four-band model with two different splitting mechanisms. In Supplementary Fig. 3(a), the splitting is solely due to the parabolic band (PB) component M 1 , while in Supplementary Fig. 3(b), it is solely due to the Zeeman effect described by the g-factor g 0 . The parameters used here are Fermi velocity v F = 9 × 10 5 m/s, M 1 = 1.9 eV nm 2 (or equivalently g ef f = 110), g 0 = 0 for Supplementary Fig. 3(a), and v F = 7 × 10 5 m/s, M 1 = 0 eV nm 2 , g 0 = −110 for Supplementary Fig. 3(b). Even though the Fermi velocities are slightly different in the two cases, we see that at low fields (B < 1 T), both mechanisms give equally well fits to the k · p LLs. However, when the magnetic field increases, g 0 fails to produce the correct LL dispersion, while the M 1 contribution still maintains reasonable agreement. From Eq. (3) of the main text, we can see that the main difference between the two models is from the field induced gap term (M B , which is related to M 1 ). Therefore, we can conclude that the M 1 parameter is primarily responsible for the (spin) splitting of the LLs in InAsSb/InSb SLs.

SUPPLEMENTARY NOTE 5. EFFECTIVE g-FACTOR ERROR ESTIMATION
In this work, we determined the g-factor error bars ( Fig. 6(b) of the main text) through the following steps.
First, we extract the LL transition energies from the magneto-absorption spectra. Supplementary Fig. 4 shows a few raw spectra after normalizing to the zero field. The LL transitions can be identified as absorption dips in the spectra with a Lorentzian lineshape.
The black dash lines are multi-Lorentzian fits to the data at low energies. The fits capture the spectral lineshape of interband LL transitions well (modes 3, 4, 5 . . . ), with a typical error as small as ±0.2 meV for each mode. Mode 2 exhibits an unusual broadening. But nevertheless, one can determine its central energy with great accuracy using Lorentzian fitting. The fit to mode 1, which is a CR mode, is affected by the presence of two strong modes at low energies (gray areas) whose energy position is independent of magnetic field.
However, once we determine all the interband LL transitions, the energy of this CR mode is known and can be deduced from the LL fan diagram shown in Fig. 3(b,d) of the main text.
Second, we recognize that in our k.p fitting, the main fitting parameter is the Kane energy E p . When using E p = 18 eV ( Fig. 3(a) of the main text), the calculated LL transitions pass the centers of most interband absorption modes and exhibit a good fit. In Supplementary   Fig. 5, we examine the robustness of the fitting by manually increasing or decreasing E p to E p = 19 eV (dash line) or E p = 17 eV (solid line) while keeping all the other parameters of sample 1445 unchanged. We find that both the dash and solid lines are not a good fit to the centers of interband LL transitions but reasonably capture the boundaries (or broadening) of the transitions. Therefore, we conclude that the error bar of E p for the k.p fitting to the sample 1445 data is ±1 eV.
Third, given the above error bar, we calculate the corresponding LLs using E p = 17 eV and E p = 19 eV and extract the g-factors in the low field regime (B ≤ 1 T). This way, we can define the upper and lower bound of the g-factors. The results are shown in Fig. 6(b) of the main text.

SUPPLEMENTARY NOTE 6. ORIGIN OF THE BAND SPLITTING
In this section, we discuss the origin of the M 1 parameter in the four-band model and its connection to the k · p model. Due to the large quantization energy in our SLs, we can follow Ref. [15] and treat the in-plane dispersion as perturbation, that is H(k x , k y , k z ) = H 0 (k z ) + ∆H(k x , k y ).
In this way, the wavefunction along the z-direction |i z can be determined by considering the eigenvalue problem H 0 |i z = E i |i z under proper boundary conditions while the in-plane wavefunction is still plane-wave-like due to its translational symmetry. Here, the eigenvalue E i corresponds to the band edges of different subbands.
Next, we can construct the four-band effective Hamiltonian H in the subspace of |1 ,|2 ,|3 ,|4 , which corresponds to the conduction and heavy-hole bands with up and down spins. Using degenerate perturbation theory [16], the Hamiltonian H close to Γ point reads H ij = i| H 0 |j + i| ∆H |j + 1 2 l i| ∆H |l l| ∆H |j 1 where (i, j) and l denote the states inside and outside the subspace, respectively. Then, one can express the M 1 parameter of the ith band as This expression involves all the components to understand the LL splitting effect. First, the LL (spin) splitting is due to the interactions with other bands outside the subspace.
It is consistent with the expectation that the LLs are strictly spin-degenerate in a pure four-band model, and any splitting ought to come from contributions of additional bands.
Second, the M i 1 is related to the sum of the matrix elements i| ∆H il |l between different bands. Therefore, the wavefunction overlap and mixing can inevitably change the values of the matrix elements and are responsible for the g-factor engineering in our system.